Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem

We investigate singular and degenerate behavior of solutions of the unstable free boundary problemΔu=−χ{u>0}. First, we construct a solution that is not of class C1,1 and whose free boundary consists of four arcs meeting in a cross-shaped singularity. This solution is completely unstable/repulsive from above and below which would make it hard to get by the usual methods, and even numerics is nontrivial. We also show existence of a degenerate solution. This answers two of the open questions in the paper [R. Monneau, G.S. Weiss, An unstable elliptic free boundary problem arising in solid combustion, http://arxiv.org/abs/math.AP/0507315, submitted for publication].